CALCULATE

CALCULATE

** Time to Double the Money calculator** uses interest rate and calculates a number of years and/or mounts needed to get money doubled based on the given interest rate. It's an online investment returns calculation tool programmed to estimate how many years and/or months are required to get your money or investment double the value.

It is necessary to follow the next steps:

- Enter a value for intereset rate in the box. This value must be positive real number;
- Press the "CALCULATE" button to make the computation;
- Time to double the money calculator will give the number of years and/or months needed to double the money.

We will use __the Rule of $72$__ to find how long it will take to double money at a given interest rate. If we divide $72$ by the interest rate we get the period needed to double money. Also, using this rule we can calculate the necessary interest rate for doubling our money within a certain time period.

For example, if we want to double money in $3$ years, we will divide $72$ by $3$ to get $24\%$ interest rate annually. The Rule of $72$ is a version of the compound interest calculation. The Rule of $72$ can be expressed by the following formula
$$R \times t = 72$$
where $R$ is interest rate per period in a percent and $t$ is number of periods.
From the previous formula, if we known interest rate, \underline{the number of periods to double money} is determined by the formula
$$t =\frac {72} R$$
If we known number of periods, the interest rate required to double money is determined by the formula
$$R = \frac{72}{t}$$
As we know, the yearly compound interest formula is determined by the formula:
$$C = P(1 + r)^t$$
where $C$ is the compound interest, $P$ is the initial principal sum of money, $r$ is the interest rate per period in decimal form, and $t$ is the time periods in years.

If we substitute $P$ by $2P$, since we want to double money, the previous formula becomes
$$2P = P(1 + r)^t$$
After dividing both sides by $P$, we obtain
$$ 2=(1 + r)^t$$
After taking the natural logarithm of both sides, we get
$$\ln 2=t\times \ln(1+r)$$ Hence, the solution for $t$ is
$$t=\frac{\ln 2}{\ln(1+r)}\approx\frac{72}r$$

Time to double the money can be applied to any problem with growing compounded rate. For example, in studying a population, GDP growth, investment returns or fees, etc. Many people, before invest the money, want to know returns, fees and time of period required to get their money double the value.

**Practice Problem 1:**

If we invest a sum of money in a bank at $5\%$ interest per year, how many years and months are required to get our money double the value?

**Practice Problem 2:**

If we invest a sum of money in a bank at $0.5\%$ interest per month, how many years and months are required to get our money double the value?

Time to double the money calculator, example calculation and practice problems would be very useful for grade school students of K-12 education to understand the concept of compound interest. It will help them to make their financial analysis, i.e. to evaluate their business, project, budget and other finance-related entities.